[seqfan] Re: Conjecture about A127750

[Allan Wechsler]

Two responses:

First, to Jean-Paul Allouche: I looked up a Hilbert/Cauchy determinant
explanation, and I think it's unfortunately not applicable in this case,
though I can absolutely see why you thought of it. In this case, the
original infinite matrix M is lower-triangular, so the inverse M' is also.
That means that the "determinant", insofar as the concept applies to
infinite matrices, is just the product of the main diagonal entries. In the
case of M, that product is 1 * (1/3) * (1/5) * (1/7) * ..., and the product
converges rapidly to 0. Similarly, the determinant of M' is the product of
the main diagonal entries, which is 1 * 3 * 5 * 7 * ... which diverges
rapidly. So the whole matrix M doesn't really have a well-defined
determinant, though it is invertible.

Second, to Robert Israel and Neil Sloane: My skills as a LaTician have been
rusting for more than 30 years. Unless I could get a co-author, I think I'm
going to have to stick with a text file uploaded to OEIS. (Probably it
would be most appropriate to co-author with Paul Barry, who first proposed
the sequence, and I have sent him email with a preliminary query. But I
don't know if he's still interested in this sequence 13 years later.)

Thanks to everybody; I'm going to crawl off and write now.

On Tue, Feb 9, 2021 at 12:45 PM <israel at math.ubc.ca> wrote:

> You might upload a text file, or even better a nicely formatted .pdf
> (produced from Latex, for example), in the Links section of the sequence.
> If you want a more formal publication, you might send it to the Journal of
> Integer Sequences, and then include a link to that with the sequence
> (first
> to the ArXiv preprint, and eventually to the published article).
>
> Cheers,
> Robert
>
> On Feb 9 2021, Allan Wechsler wrote:
>
> >I'm afraid I'm not familiar with Hilbert determinants and Cauchy
> >determinants -- but I do have a proof of the stated conjecture, and am
> just
> >wondering what to do with it. Even if stated tersely, it wouldn't fit
> >comfortably in the sequence comments. Do I need to publish a brief paper
> to
> >ArXiv and reference it? Or should I upload a text file to OEIS itself? My
> >main theorem is that A127750(n+1) = 2 * A001151(N) - A209229(N), and the
> >conjecture follows as an easy corollary.
> >
> >On Tue, Feb 9, 2021 at 12:41 AM Jean-Paul Allouche <
> >jean-paul.allouche at imj-prg.fr> wrote:
> >
> >> Hi
> >>
> >> Is it conceivable that this determinant has an explicit form (à la
> >> Hilbert determinant or à la Cauchy determinant)? best jean-paul
> >>
> >> Le mar. 9 févr. 2021 à 06:25, Allan Wechsler <acwacw at gmail.com> a
> écrit :
> >>
> >> > Recently I was investigating a combinatorial system (a simple Turing
> >> > machine, in fact), and became curious about a sequence it was
> >> displaying. I
> >> > looked up the sequence on OEIS and found https://oeis.org/A127750,
> >> > which matched perfectly.
> >> >
> >> > The description in the entry had absolutely nothing to do with my
> >> > generating system. But there is a conjecture, apparently due to either
> >> the
> >> > author, Dr. Paul Barry, or to Neil Sloane -- the entry isn't entirely
> >> > clear.
> >> >
> >> > Obviously I wanted to know if my sequence and Barry's were the same. I
> >> was
> >> > able to analyze my own sequence fairly easily, and the conjecture was
> >> quite
> >> > trivially true for my sequence. If I could prove the identity of the
> >> > two sequences, I would have proven the conjecture.
> >> >
> >> > Well, now I think I have proven it. What should I do next?
> >> >
> >> > --
> >> > Seqfan Mailing list - http://list.seqfan.eu/
> >> >
> >>
> >> --
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> >>
> >
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